Environmental Engineering Reference
In-Depth Information
ʴ
J
i
(ˆ)
ʩ
i
with
that relates a variation
in the mean concentration of nutrients in
0
in the emission rates
Q
j
and initial distribution of nutrient
ʴ
ʴˆ
variations
Q
j
and
0
. It makes the estimates (
2.25
) and (
2.35
) rather efficient and computationally
economical, because the solutions
g
i
(
ˆ
r
j
,
t
)
, once found, can be re-used in these
0
formulas for different values of
Q
j
,
r
j
or
.
The effect of changing the position of sources from
r
j
to
r
´
j
,
j
ˆ
(
r
)
=
1
,...,
N
,is
estimated by the formula
T
g
i
(
Q
j
(
N
r
´
j
,
ʴ
J
i
(ˆ)
=
)
−
g
i
(
r
j
,
)
)
.
t
t
t
dt
(2.36)
0
j
=
1
Finally, we give without proof a general sensitivity formula
T
N
0
ʴ
J
i
(ˆ)
=
g
i
(
r
j
,
t
)ʴ
Q
j
(
t
)
dt
+
g
i
(
r
,
0
)ʴˆ
(
r
)
dr
0
D
j
=
1
T
T
−
g
i
(
,
)ˆ(
,
)ʴʶ(
,
)
−
g
i
(
,
)
(
,
)
,
r
t
r
t
r
t
dSdt
r
t
B
r
t
drdt
0
S
T
0
D
(2.37)
where
v
s
∂ˆ
∂
B
(
r
,
t
)
=
ʴ
U
·∇
ˆ
−∇·
ʴμ
∇
ˆ
+
ʴ˃ˆ
+
ʴ
z
,
0
cf. [
43
], taking into account arbitrary variations
ʴ
Q
j
(
t
)
and
ʴˆ
(
r
)
, and small varia-
tions
in the domain
D
. Unlike the previous formulas, estimate
(
2.37
) is more complicated, because it uses solutions of both problems (
2.4
)-(
2.11
)
and (
2.18
)-(
2.24
) and linearised equations for perturbations.
ʴ
U
,
ʴ˃
,
ʴμ
,
ʴ
v
s
and
ʴʶ
2.5 Main and Adjoint Numerical Schemes of the Dispersion
Problem
In this section, balanced and absolutely stable second-order finite diference schemes
based on the application of the splittingmethod byMarchuk [
22
] and Crank-Nicolson
schemes [
8
] are developed to solve numerically the dispersion model (
2.4
)-(
2.11
)
and its adjoint formulation (
2.18
)-(
2.24
). Since they were described in detail in Skiba
[
41
], we give here only basic results.
Using the continuity Eq. (
2.11
), the operator
A
of Eq. (
2.4
) can be written as
A
=
A
1
+
A
2
+
A
3
, where
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